# Lesson Explainer: Applications of Triangle Congruence Mathematics • 8th Grade

In this explainer, we will learn how to use the triangle congruence criteria SSS, SAS, ASA, and RHS to find unknown angles or sides in geometry problems.

We can recall that there are many different ways of determining whether two triangles are congruent. For example, we can check whether they have three congruent sides, have two congruent sides and the included angles are congruent, or have two congruent angles and the shared sides are congruent. These are called the SSS, SAS, and ASA congruence criteria respectively.

Finally, there is an extra congruence criterion for right triangles called the RHS criterion, which says that any two right triangles with congruent hypotenuses and any congruent side must be congruent.

We can use these congruence criteria to determine if two triangles are congruent, which shows that their corresponding sides and angles are congruent. This means we can find unknown angles and sides by using triangle congruency.

Letβs see an example of using congruent triangles to prove a geometric property by considering the diagonals of a parallelogram. We recall that a parallelogram is a quadrilateral with opposite sides that are parallel and have the same length. We can note that one diagonal of this parallelogram splits the shape into two triangles: and .

We can see that both triangles have three congruent sides:

Hence, the triangles are congruent by the SSS criterion. Therefore, the corresponding angles in the triangles are congruent, so and .

We can do the same with the other diagonal as shown.

If we sketch both diagonals on the parallelogram and label the point of intersection between the diagonals , then we can note that triangles and have two congruent angles and the included side is congruent.

Thus, they are congruent by the ASA criterion. This means that their corresponding sides are congruent, so and .

Hence, we have shown that the diagonals of a parallelogram bisect each other.

Letβs now see another example of showing a useful geometric property by using triangle congruence.

### Example 1: Applying Properties of Congruence to Solve Problems

In the figure, and are congruent.

1. Work out the length of .
2. Work out the length of .
3. Work out measure of angle .

We start by recalling that we say that two polygons are congruent if their corresponding side lengths and angles are congruent. Since we are told that triangles and are congruent, we can conclude that their corresponding sides have the same length and their corresponding angles have the same measure.

In congruence, the order we write the vertices in tells us the corresponding sides and angles. We can also see this in the diagram.

We have

Part 1

Using the congruence of the two triangles, we know that is the same length as . Since we must have

Part 2

Using the congruence of the two triangles, we know that is the same length as . Since we must have

Part 3

Using the congruence of the two triangles, we know that the interior angle at must have equal measure to the interior angle at since these are corresponding angles.

Since we must also have

In our next example, we will use the congruency of triangles to find the measure of an angle in a given diagram.

### Example 2: Finding the Measure of an Angle in a Triangle Using RHS Congruence

In the following figure, find .

We note that the diagram contains two right triangles: and . We can also see that and these triangles share side .

We recall that the RHS criterion for triangle congruency tells us that if right triangles have congruent hypotenuses and one side congruent, then they are congruent. Hence, by the RHS criterion.

This means that their corresponding angles and side lengths are congruent. We note that corresponds to , so both angles have the same measure.

Hence,

In our next example, we will find the lengths of multiple sides in a geometric construction by using triangle congruence.

### Example 3: Finding a Side Length in a Triangle Using ASA Congruence

Find the length of and .

We first note that we are given the measures of two interior angles of each triangle and that these measures are equal:

Similarly, we can notice that the included sides of these angles are equal in length since

So, the two triangles have two angles of equal measure, and the included sides are congruent. Hence, by the ASA congruence criterion, we have .

This means that their corresponding sides and angles must be congruent. We see that side corresponds to . So, these sides are the same length.

Thus, .

We also note that corresponds to , so these line segments have the same length. Adding this length and the length of onto the diagram gives us the following.

We see that . Hence,

Therefore, and .

In our next example, we will determine the measure of an angle by using the congruency of triangles.

### Example 4: Finding the Measure of an Internal Angle Using SAS Congruence and Complementary Angles

Given that is a square, find .

We first note that we are given two congruent line segments: and . We can use this and the fact that is a square to show that triangles and are congruent.

Since is a square, we can add right angles at the vertices of the square. Similarly, the sides of a square are all the same length, so . This gives us the following.

We see that triangles and are both right triangles with congruent legs. In particular, we have

This means that they have congruent sides and congruent included angles, so they are congruent by the SAS criterion.

We note that since it is the interior angle of a square. We can determine by noting it adds to to give a right angle. So,

We can add this onto the diagram.

Since , their corresponding angles are congruent. We note that corresponds to , so their measures are equal.

Hence,

In our final example, we will find missing lengths in a kite by using triangle congruence.

### Example 5: Finding Missing Lengths by Applying ASA Congruence

Find the lengths of and .

We begin by noting that we are given pairs of congruent angles in the diagram:

Since the included side of these angles is shared between triangles and , we know that is a common side and we can show that these triangles are congruent.

By the ASA criterion, we can conclude that . This means that their corresponding sides and angles must all be congruent. We can highlight the corresponding sides on the diagram.

We have , so . Similarly, we have , so .

Letβs finish by recapping some of the important points from this explainer.

### Key Points

• We can use the congruency of triangles to prove geometric properties such as the property of the diagonals of a parallelogram bisecting each other.
• We can use the congruency of triangles to determine the lengths of missing sides and the measures of missing angles in some geometric constructions with triangles.